Use, Mention, and Identity

Ed plausibly maintains that the following argument is invalid:

Hesperus is so-called because it appears in the evening
Hesperus = Phosphorus
--------------
Phosphorus is so-called because it appears in the evening.

But then he asks: if the above is invalid why isn't the following argument also invalid?

'Hesperus’ designates Hesperus
Hesperus = Phosphorus
-------------
‘Hesperus’ designates Phosphorus.

I say both arguments are valid. The second strikes me as obviously valid.  As for the first, suppose we rewrite it by replacing 'so-called' with an equivalent expression. We get an argument I will call the REWRITE:

Hesperus is called 'Hesperus' because it appears in the evening
Hesperus = Phosphorus
-------------
Phosphorus is called 'Hesperus' because it appears in the evening.

Now the conclusion of the REWRITE is admittedly strange. But it is true! Phosphorus is called 'Hesperus' when it appears in the evening, and it is called that because it appears in the evening.   So the REWRITE is valid, whence  it follows that the first argument, pace Ed, is valid.

So both arguments are valid.    

UPDATE (9/12). My thesis is refuted in the combox. But as Chisholm once said after some point of his had been refuted, "Well, at least I said something clear enough to be refuted!"  I am not suggesting, however, that Ed's suggestion that the second argument supra is invalid has any merit.

Comments

[The London Ostrich]

>> As for the first, suppose we rewrite it by replacing 'so-called' with an equivalent expression. [My emphasis]
Nice try, but the expression, ‘so called’, while equivalent in the major, is not equivalent in the conclusion.

Phosphorus is so-called because it appears in the evening.

means

Phosphorus is called ‘Phosphorus’ because it appears in the evening.

not

Phosphorus is called 'Hesperus' because it appears in the evening.

The first argument is invalid because of the equivocation on ‘so-called’. So I repeat, why might there not be a similar equivocation in the ‘designates’ argument?

[Valeriu]

REWRITE is valid, but it is not the correct rewriting of the first argument. The phrase 'so-called' should be replaced differently in the premise and in the conclusion.

REWRITE 1:

Hesperus is called 'Hesperus' because it appears in the evening
Hesperus = Phosphorus
-------------
Phosphorus is called 'Phosphorus' because it appears in the evening.

This is not a valid argument.

[Valeriu]

Sorry for my inattentiveness to The London Ostrich's comment. It says the same thing.

[BV]

And what is the equivocation in the second argument? Is it on 'designates'?

[The London Ostrich]

>>Sorry for my inattentiveness to The London Ostrich's comment. It says the same thing.

And I agree with Valeriu.

>>And what is the equivocation in the second argument? Is it on 'designates'?

Yes. Or rather on the construction 'designates X'

[BV]

You need to tell us what the nature of the equivocation is. What meaning does 'designate' have in the major, and what different meaning does it have in the conclusion?